Noise impact assessment

In ISC, the idea of torque control is the same as in conventional hysteresis DTC [TN1986]

or DSC [DE1988], i.e. controlling the angle between the stator- and the rotor-flux vector (δT in Fig. 5.2), while the stator-flux magnitude is regulated by means of feed-forward voltage terms. The shown control scheme can be realized in the stationary or in the synchronously rotating reference frame. In this contribution, the realization is based on the stator-fixed reference frame. The analysis and the design of the control system in both cases of reference frames remain more or less the same.

Control principle: The stator voltage can be represented in stator coordinates as the summation of time-derivative of flux and ohmic stator voltage drop,

u_s= ˙ψ_s+ i_sRs. (5.9)

The electromagnetic torque is the cross product of rotor permanent-flux vector ψ_p and the stator-current vector i_s. It can also be represented as given in (5.10).

T = 3

2p^hi_s× ψ_pⁱ= 3 2p

" ψ_s− ψ_p Ls

!

× ψ_p

#

= 3 2

p Ls

[ψpψssin(δT)] ∝ sin(δT)

(5.10)

where ψs and ψp are the stator- and rotor-flux magnitudes. Equation (5.10) implies that the torque can be controlled by the angle between stator- and rotor-flux vector directly,

-T^∗

Tˆ

u^∗_s δ^∗_T

ψˆ

s ǫ

Voltage vector generation

Observer

Motor PWM

ˆ

x = estimate(x)

Figure 5.27: ISC structure

5.2 Indirect Stator-quantity Control 91

with the assumption of constant flux magnitudes. Therefore, for a given torque reference T^∗, the required reference torque angle δ_T^∗ can be generated using (5.10). With known rotor-flux position ǫ and the command of torque angle δ_T^∗, the required stator-flux position (θ^∗_s = ǫ+δ_T^∗) can be located. With the help of (5.9), the required voltage vector to bring the stator-flux vector from its present location to the required location can be evaluated. The stator-flux magnitude reference is set depending on the torque command. This is because the stator-flux is the vectorial sum of the rotor-flux vector and the part contributed by the stator-current vector via the armature inductance Ls. The details of above explanation can be understood more clearly with Fig. 5.28 which also gives an idea how the difference equation can be realized in a sampled system.

Control design: The PWM with zero-sequence involved in the control loop, it is pos-sible here also to utilize asymmetrical regular sampling for the controller design with 5 kHz switching frequency and 10 kHz controller sampling frequency. The control loop is divided into two parts in a cascaded fashion, the inner flux control and the outer torque control shown in Fig. 5.29. In the original ISC described in [UB1989], the predictive slip compensation is performed in the torque control which indeed make the torque and the flux control function parallel instead of cascade. In the further discussion we would like to continue with the cascaded structure.

A very important note for the design is that the dead-time due to computation is neces-sarily to be considered in the innermost loop. The control and the plant model together can be represented as shown in Fig. 5.29 for better and easy understanding. The block schematic in this figure gives a clear connection between the control and the plant modules and helps in simplification of loops.

Flux control ler: The assumption of exact compensation of the stator-resistance drop on the controller side in Fig. 5.29 simplifies the flux controller as shown in Fig. 5.30. It is not to be confused with flux-vector control, here voltage reference is actually calculated in feed-forward manner as given below.

u^∗_s = ∆ψ_k

∆t , where ∆ψ_k = ψ^∗_k− ψ

k−1

As pointed out above, the dead-time in the control loop is to be considered in the inner-most loop, hence it is taken care in Fig. 5.30. In this flux loop, the time divisor ∆t^-1 (for numeric difference) actually acts as a constant gain and it can be defined as K.

Figure 5.28: ISC voltage vector generation

-- Motor Flux feed-forward controllerTorque controller

- Controller

-- T∗ T

T TNKpN

ˆ ψ ^s

isRsisRs

u∗ sus δ∗ TPolartoCartesian topolarcartesian ǫ ǫ

ǫ

θ∗ s

1 J

ψ∗ s

1 ∆tTΣ

TL

ψp ψs θsδTPWM ω p

ψ∗ s

Figure 5.29: ISC plant and control model

5.2 Indirect Stator-quantity Control 93

-ψ^∗

s

ψ_s

ψs

u_s K

Controller Computation

delay

e^-sTc fc

Motor

Figure 5.30: Flux-loop design

The open-loop gain for the system shown in Fig. 5.30 can be represented in s and z domain as in (5.11). The corresponding open-loop Bode plot with K=1 is shown in Fig.

5.31.

L1(s) = K s e^-sTc

"

1 − e^-sTc

s

#

L1(z) = K

"

0.0001z^-2 1 − z^-1

# (5.11)

In order to have adequate damping for the closed-loop flux response, one can choose upper end of phase margin, but that has to ensure the closed-loop response has a unity gain until the operating frequency range of the motor. The considered motor has a rated fundamental frequency of 250 Hz (1.57·10³ s^-1). Depending on the operating range, e.g., taking field-weakening into account, the control loop has to be adjusted. Keeping this in mind, a higher damping in the flux-loop does not make sense. In order not to loose much gain for the flux control, the phase margin is reduced to 57^◦. The corresponding

-100 -50 0 50

Magnitude (dB)

10^-1 10 10¹ 10² 10³ 10⁴ 10⁵

-180 -135 -90

Phase (deg)

Gm = 80 dB (at 10.5k s ) , Pm = 90 deg (at 1 s )^{-1 -1}

ω (s )^-1

Figure 5.31: Open-loop Bode plot for L1(z)

gain K for this phase margin is 3550 s^-1, i.e. approximately ∆t=280 µs. With this gain the closed-loop transfer function is found as in (5.12).

T 1(z) = 0.355z^-2

1 − z^-1+ 0.355z^-2 (5.12)

Bode diagram and step response for the transfer function (5.12) are shown in Fig. 5.32.

In the frequency plot, the closed-loop bandwidth for the flux-loop is observed as high as 8.5·10³ s^-1 or 2π·1.35 kHz. The overshoot observed in the step response is around 8%, because of non-conservative phase margin considered in the design.

Torque-control ler design: From Fig. 5.28 it is clearly visible that no additional torque controller would be necessary, provided one can construct the reference torque angle perfectly from the reference torque and the reference flux vector is regulated without any delay. If these two criteria are met, the output of the speed controller or the independent torque reference can directly be used to drive the flux control. But that is not truly the case, because the flux-loop dynamics is finite and therefore cannot ensure the proper regulation of the reference angle. Hence there always exists a difference between the actual and the reference angle. Also the quality of torque regulation is highly dependent on the construction of the reference torque angle. With ideal assumptions it may work, but usually in practice to ensure reliable torque regulation an additional torque-controller is preferred at the cost of reduced closed-loop torque dynamics.

For design, the compensation of gain factors such as stator- and rotor-flux magnitudes in the controller is assumed to be exact as of the plant values. With these assumptions the plant for the torque-controller design is simply T 1(z) which is given in (5.12). By looking at the Bode plot (Fig. 5.32) and the transfer function T 1(z), it seems that a simple PI-controller with an appropriate time constant will meet the performance requirements.

The reset time constant of the PI-controller is chosen same as the corner frequency of

-20 -15 -10 -5 0 5

Magnitude (dB)

-450 -360 -270 -180 -90 0

Phase (deg)

0 0.5 1 1.5 2 2.5

x 10^-3 0

0.2 0.4 0.6 0.8 1

Time (s)

Amplitude

ω (s )^-1

10⁴ 10⁵

10³ 10²

Figure 5.32: Frequency and time response of closed-loop flux-controller

5.2 Indirect Stator-quantity Control 95

the double pole of the flux-loop transfer function T 1(z). Then, the open-loop transfer function for the complete torque loop can be given as in (5.13). To avoid any additional sampling delay, the integrator of the torque PI-controller is discretized using the bilinear or Tustin’s approximation. It is necessary because computation delay and sampling are already included in the design of the flux control.

L2(z) =

"

KpN+ KpN

T_N Tc

2

1 + z^-1 1 − z^-1)

!#

T 1(z) (5.13)

With KpN=1 and TN=₈·^1013s-1=125 µs, the open-loop Bode response is shown in Fig. 5.33.

In order to achieve a good compromise between closed-loop bandwidth and damping, the phase margin is chosen as 55^◦, while the corresponding gain is KpN=0.31. With this gain, the expected closed-loop response in frequency and time domain is shown in Fig. 5.34. The closed-loop torque bandwidth is around 6.5·10³s^-1 and the overshoot observed in the step response is close to 6%. The bandwidth is almost the same as that of sampled FOC but the overshoot is higher in comparison. It is understandable, because the considered ISC structure has a cascade of two controllers, while FOC has independent single controllers.

5.2.2 Quasi-continuous control design

A detailed introduction of the quasi-continuous control has already been given in section 5.1.2. To design the controller, it is required to consider the similar criteria defined for FOC, due to the presence of PWM in the loop. Unlike FOC, in the proposed ISC the torque is controlled via the flux-control loop in a cascaded fashion. Similar to the above sampled control, here also controllers are considered one after the other for the design procedure. Quasi-continuous FOC design has focused mainly on the current ripple introduced by the PWM voltage at the motor terminals. However, ISC does not have any particular controllers which act on the current, instead the controllers acting on the

-20 0 20 40 60

Magnitude (dB)

-450 -360 -270 -180 -90 0 90

Phase (deg) Flux loop T1(z)

Torque Controller Torque Open-loop

ω (s )^-1

10² 10³ 10⁴ 10⁵

Figure 5.33: Bode plot for open-loop torque transfer function

-30

Figure 5.34: Frequency and time response of closed-loop torque control

torque and flux have ripple introduced indirectly by the PWM voltage. In order to make the analysis simpler, the torque and the flux ripple (which will be used in the design procedure) can be defined as a function of the current ripple as in (5.14) and (5.15), respectively: Control design: The flux-controller gain K calculated in the previous section was auto-matically restricted by the controller dead-time. Due to the neglection of this dead-time (because it is too small) in the quasi-continuous approach, the inner flux loop turns out to be a simple first-order delay. Ideally the gain for K is infinite, but to have flexibility in the torque-controller design one can fix the gain of the flux loop such that it gives just the adequate performance to regulate the alternating flux references. The rated frequency of the motor is 250 Hz. With field-weakening considered the flux loop is designed such that it gives a bandwidth around 1 kHz. The corresponding gain is

K = 1

150 µs = 6666.66 s^-1.

To make the design approach further simpler, the torque-controller time constant TN is selected as the time constant of the flux-loop, i.e. 150 µs.

The slope and the peak-peak ripple of the reference voltages are the important factors for control design. Both these factors are finally decided in the flux controller. Ripple and slope of the reference and the actual signal at the input of the flux control have to be evaluated, to find finally the factors for the reference voltages. Unlike in FOC where the reference currents were assumed as free of ripple and slope, in this case the input flux reference cannot be considered as ripple-free, because it is the output of the torque controller. The proportional and the integral gains of the torque controller act on the actual torque ripple (ref. (5.14)) and the ripple carries this further in the reference torque

5.2 Indirect Stator-quantity Control 97

angle δ_T^∗. Hence a ripple will be seen in both α- and β-flux references. If the mean value of the torque angle is δT, the ripple is defined as ∆δT. This ripple can be evaluated from (5.16) using (5.10); the ripple of the flux reference has the opposite sign of the torque ripple due to the subtraction at the input of the torque controller.

∆δT ≈ −2 3

Ls

pψ_pψ_s∆T (5.16)

Slope calculation: The slope of ripple which appears before the flux-controller gain K is the sum of the slopes contributed by the reference and the actual flux components. The reference slope is a function of the actual torque ripple slope and the torque PI-controller gain. In case of FOC the slope enhancement of the integral part of the controller was completely neglected because of its high time constant. But here in ISC one cannot neglect the integral gain in the torque-controller because the integral time constant is much smaller in comparison and hence contributes to the slope. The steady-state representation of the torque ripple and correspondingly the slopes from the proportional (P: in black) and integral channel (I: in blue) and from both together (PI: in red) of the torque PI-controller are represented in Fig. 5.35. In the figure Ton and Toff correspond to increasing and decreasing torque-ripple timings, respectively. For a particular operating point they are evaluated using (5.17), wherein the idea is to consider the maximum slope, hence the applied q-axis voltage taken as ^2u₃^dc.

Ton = 2 3p

∆T ψp

Ls

2udc

3 − ωψ_p^ Toff = 0.5Ts− Ton.

(5.17)

Further, the slope of the torque-controller output can be evaluated by using (5.18) and (5.19).

Slope (,,)PIPI

t

t

∆T

∆T

Ton

Toff

Ts

2

Figure 5.35: Slope of torque-controller components

Slope during Ton:

SP Ton = −KpN

ψs

2udc

3 − ωψp



S_ITon =

Z S_{P Ton} TN

dt

(5.18)

Slope during Toff:

S_{P Toff} = −S_ITonTon

Toff

SIToff =

Z SP Toff

TN

dt

(5.19)

where the slopes during on and off time for P - and I-part of the controller are repre-sented as SP Ton, SP Toff and SITon, SIToff, respectively. To solve (5.18) and (5.19), a simple numerical method can be used. The timings seen in Fig. 5.35 are arbitrary, they change with the operating point (ω), which is well taken care by (5.17).

In the control scheme, the reference torque angle δ^∗_T is added to the rotor-flux position ǫ to find the reference stator-flux-vector position θ^∗_s and with known flux-reference magnitude ψ_s^∗ one can estimate the reference flux vector in stator coordinates. With the help of Polar to Cartesian transformation of this flux vector one can find the ψ^∗_sαand ψ_sβ^∗ flux-reference components. Using the already calculated ripple slope from (5.18) and (5.19), the slope for these reference fluxes can be found; the procedure is briefed below. In steady-state operation θ^∗_s can be presented with the mean and the ripple of the reference torque angle.

θ_s^∗ = θs+ ∆δ^∗_T =^hǫ + δT

i+ ∆δ^∗_T

The reference flux component can be evaluated with the help of the transformation.

ψ^∗_sα= ψ_s^∗cos(θ_s^∗) = ψ_s^∗cos(θs+ ∆δ^∗_T)

To calculate the slope, the time derivative of the above equation is essential.

ψ˙^∗_sα = ψ_s^∗[−sin(θ_s+ ∆δ^∗_T)] [ ˙θ_s+ ∆ ˙δ^∗_T]

≈ ψ_s^∗[−sin(θs)] [ ˙θs+ ∆ ˙δ^∗_T]

≈ ψ_s^∗[−sin(θ_s)] [ω + ∆ ˙δ^∗_T

| {z }

]

It is evident from the above equations that the slope depends on the speed and the slope of the reference torque-angle ripple; under-braced term (calculated using (5.18) and (5.19)).

Due to the negation of torque slope at the torque-controller summation, one will see a reduction in the peak value of slope in the above equation as the motor speed increases.

The total slope which appears at the output of the flux-controller summation is the instantaneous addition of reference and actual flux slopes. For simple understanding, the slopes of these two signals just before the gain K of the flux-controller can be represented as in Fig 5.36. The bold blue line shows the maximum and the minimum limits of slope

5.2 Indirect Stator-quantity Control 99

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04

-400 Slope band of actual flux

Slope of reference flux components

(a) Estimated reference and actual flux-signal slopes at 25 Hz (^f_f^s=200)

0 0.5 1 1.5 2 2.5 3 3.5 4

Slope band of actual flux

Slope of reference flux components

Operating with f= 250 Hz (K_pN=1 T_N=150 µs)

(b) Estimated reference and actual flux-signal slopes at 250 Hz (^f_f^s=20)

Figure 5.36: Estimated slopes for reference and actual flux component at 25 Hz and 250 Hz (Rated fundamental frequency=250 Hz)

for the actual flux: The shape resembles the applied phase-to-neutral voltage and the maximum value corresponds to ^u_1.5^dc. The maximum slope of the control error (ψ_sα^∗ − ψ_sα and ψ^∗_sβ− ψsβ) can be evaluated by adding the slopes shown in the figure. As the slope of the actual flux cannot be controlled, the only possibility is in controlling the reference flux, because it is a function of torque-controller parameters. The maximum allowable slope of the flux-controller error (SF Err) can be calculated back from the maximum allowable slope of the reference voltage (SCarrier) as given in (5.20).

SF Err = SCarrier

Based on (5.20), the torque-controller gain has to be adjusted such that the reference-voltage slope will never go beyond the maximum limit. While satisfying this criterion, the additional slope incorporated by the zero-sequence injection in PWM should also be taken into consideration. The evaluated gain K_pN for the torque-controller which satisfies the slope constraint needs to be cross-verified for the other criterion, i.e. maximum allowable peak-to-peak ripple in the reference voltages.

Peak-to-peak ripple calculation: The maximum ripple in flux difference appears at the peak of the reference voltage. The peak-to-peak stator-flux ripple magnitude can be defined as

The ripple given in (5.21) has two parts, the first part is contributed by the ripple-current vector ∆i_s via the stator-armature inductance and the second part is also due to the same factors through the torque control, wherein it is modulated by the controller gains. The ripple in vector ∆i_s is calculated from (5.7) as a function of modulation index. Similar to the slope calculation, here also the integrator gain for the ripple contribution from the reference flux vector has been taken into account. Therefore in (5.21) the additional variable Ki is defined as approximated integral gain for the current-ripple frequency i.e.

2fs. The gain can be calculated from the simple transfer function of the integrator, i.e.

Ki = |K

jω| = K 2π2f_s

= 6666.66 s^-1

2π10·10³s^-1 = 0.11 .

Based on the available voltage margin and with the help of (5.21) it is possible to evaluate the maximum possible torque-controller gain KpN as function of the modulation index M.

The allowable maximum gain KpN which can be used in the torque controller satisfying both criteria can be plotted as shown in Fig. 5.37. Graphs with different colors represent the gains with the respective criterion: Red and blue colored plots for the gain values vs. M for the criteria 1 and 2, respectively. In order to satisfy both criteria, one has to choose the smaller value represented as dashed black line. The possible bandwidth of the controller is also shown in Fig. 5.37.

The achievable bandwidth at lower speeds may not be too high, compared to the sampled controller. It has to be remembered that the phase margin of the torque control in the whole range is still 90^◦ because the closed torque-loop has been reduced to a simple first-order delay. Similar to FOC, here also the phase margin can be exploited to enhance the performance. A more practical feedback delay can be introduced in the current measure-ment system. For the sake of simplicity, let us assume that both feedback paths have the same delay characteristic. With the smallest gain KpN=1 from Fig. 5.37 corresponding

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

0 1 2 3 4

Gain from ripple criterion Gain from slope criterion Allowable gain from both criteria

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

0 1000 2000 3000

Bandwidth (Hz)

M Maximum allowable value ofKpN

Figure 5.37: Maximum torque-controller gain K_pN and closed-loop bandwidth as function of modulation index

5.2 Indirect Stator-quantity Control 101

to the modulation index M = 0, the bandwidth enhancement as a function of the feed-back first-order delay is shown in Fig. 5.38. From the figure, it is visible that with the introduced delay in feedback, the control response has been enhanced considerably. It is always possible to make the system performance better by designing appropriate delays for the individual loops instead of choosing the same value. It is more logical to use the feedback-filter characteristics to reduce the ripple magnitude and the slope as it further eases enhancing the controller gains and bandwidth for control.

With regard to comparison with the quasi-continuous FOC, the ISC is still behind. All depends on tuning the controller carefully as it is not too difficult to achieve the per-formance of FOC. The tuning of the controller refers to adjustment in the gains of the controllers, depending on the feedback-filter characteristics. E.g. if the filter is able to reduce the peak-peak current ripple by 25%, then the controller gain can directly be

With regard to comparison with the quasi-continuous FOC, the ISC is still behind. All depends on tuning the controller carefully as it is not too difficult to achieve the per-formance of FOC. The tuning of the controller refers to adjustment in the gains of the controllers, depending on the feedback-filter characteristics. E.g. if the filter is able to reduce the peak-peak current ripple by 25%, then the controller gain can directly be

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